3 TIỀN MÃ HÓA TUYẾN TÍNH VÀ STBC CHO HỆ THỐNG MIMO
3.1.1 Cấu trúc của bộ lập mã
Một bộ lập mã bao gồm khối mã hóa kênh, khối xen (interleaving) và khối ánh xạ ký hiệu từ đó đưa ra các ký hiệu vector đến khối tiền mã hóa. Chúng ta phân thành hai loại cấu trúc của bộ lập mã dựa trên khối ánh xạ ký hiệu: Hợp kênh không gian và mã hóa ST. Cấu trúc của hợp kênh không gian phân kênh các bit đầu ra của khối mã hóa kênh và khối interleaving thành những dòng bit độc lập. Những dòng bit này sau đó ánh xạ vào những ký hiệu vector và đưa vào trực tiếp cho khối tiền mã hóa như hình vẽ 3.2.
the transmitter, and they can both agree on a precoding algo- rithm. The capacity of the channel with CSIT (now denoted by
U) can then be achieved by a single Gaussian codebook designed
for a channel without CSIT, provided that the code symbols are dynamically scaled by a power-allocation function determined by the CSIT C=max f E 1 2log(1+hf(U)) , (10)
where the expectation is taken over the joint distribution of h
and U. In other words, the combination of this power-allocation
function f(U)and the channel creates an effective channel, out-
side of which coding can be applied as if the transmitter had no CSIT. This insight, in fact, can be traced back to Shannon in [4]. For a scalar fading channel, therefore, the optimal use of CSIT is for temporal power allocation.
This result has been subsequently extended to the MIMO fad- ing channel [6]. Under similar assumptions, the capacity-optimal input signal with CSIT can be decomposed as the product of a codeword optimal for a channel without CSIT and a weighting matrix dependent on the CSIT. The optimal use of CSIT is now linear precoding, which allocates power in both spatial and tem- poral dimensions. In other words, the capacity-optimal signal is zero-mean Gaussian distributed with the covariance determined by means of the precoding matrix. This optimal configuration is shown in Figure 5.
These results establish important properties of capacity- optimal signaling for a fading channel with CSIT. First, it is optimal to separate the function that exploits CSIT and the
channel code, which is designed for a channel without CSIT. Second, a linear precoder is optimal for exploiting the CSIT. These separation and linearity properties are the guiding prin- ciples for MIMO frequency-flat precoder designs. In particular, this article focuses on designing a precoder based on the CSIT, given predetermined channel coding and detection technique. Before discussing about specific designs, however, the structure of a system with precoding is analyzed next.
PRECODING SYSTEM STRUCTURE
The transmitter in a system with precoding consists of an encoder and a precoder, as depicted in Figure 5. The encoder intakes data bits and performs necessary coding for error correc- tion by adding redundancy, then maps the coded bits into vector symbols. The precoder processes these symbols before transmis- sion from the antennas. At the other side, the receiver decodes the noise-corrupted received signal to recover the data bits, treating the combination of the precoder and the channel as an effective channel. The structures of these processing blocks are discussed in detail next.
ENCODING STRUCTURE
An encoder contains a channel coding and interleaving block and a symbol-mapping block, delivering vector symbols to the pre- coder. We classify two broad structures for the encoder: spatial multiplexing and ST coding, based on the symbol mapping block. The spatial multiplexing structure de-multiplexes the output bits of the channel coding and interleaving block to generate inde- pendent bit streams. These bit streams are then mapped into vec- tor symbols and fed directly into the precoder, as shown in Figure 6. Since the streams are independent with individual SNR, per-stream rate adap- tation can be used.
In ST coding structure, on the other hand, the output bits of the channel coding and interleaving block are first mapped directly into symbols. These symbols are then processed by a ST encoder (such as in [38], [39]), pro- ducing vector symbols as input to the precoder, shown in Figure 7. If the ST code is capacity lossless for a channel with no CSIT (for example, the
Alamouti code for a 2×1 channel
[38]), then this structure is also capac- ity optimal for the channel with CSIT. The ST coding structure contains a single data stream; hence, only a single rate adap- tation is necessary. The rate is controlled by the FEC-code rate and the constellation design.
The difference between these two encoding structures therefore lies in the temporal dimen- sion of the symbol-level code. Spatial multiplexing spreads symbols over the spatial dimension alone,
[FIG5] An optimal configuration for exploiting CSIT.
N W^ i.i.d. Gaussian CSIT Transmitter Precoder F Encoder C X W Channel H Y Decoder +
[FIG6] A multiplexing encoding structure.
Symbol Mapping FEC Code Interleaver Symbol Mapping Input C DEMUX bk
[FIG7] A space-time (ST) encoding structure.
ST Code FEC Code Interleaver Symbol Mapping Input C bk
IEEE SIGNAL PROCESSING MAGAZINE [92] SEPTEMBER 2007
Authorized licensed use limited to: QUEENSLAND UNIVERSITY OF TECHNOLOGY. Downloaded on July 31,2010 at 16:34:04 UTC from IEEE Xplore. Restrictions apply.
Hình 3.2: Cấu trúc mã hóa hợp kênh không gian [5, 6]
Trong cấu trúc mã hóa ST, dòng bit ra của mã hóa kênh và khối interleaving trước hết ánh xạ trực tiếp thành các ký hiệu. Những ký hiệu này sau đó được xử lý bằng mã hóa ST (như trình bày trong chương 2 là một ví dụ) tạo nên các ký hiệu vector và đưa vào bộ tiền mã hóa như hình vẽ 3.3.
42
U) can then be achieved by a single Gaussian codebook designed
for a channel without CSIT, provided that the code symbols are dynamically scaled by a power-allocation function determined by the CSIT C=max f E 1 2log(1+hf(U)) , (10)
where the expectation is taken over the joint distribution of h
and U. In other words, the combination of this power-allocation
function f(U)and the channel creates an effective channel, out-
side of which coding can be applied as if the transmitter had no CSIT. This insight, in fact, can be traced back to Shannon in [4]. For a scalar fading channel, therefore, the optimal use of CSIT is for temporal power allocation.
This result has been subsequently extended to the MIMO fad- ing channel [6]. Under similar assumptions, the capacity-optimal input signal with CSIT can be decomposed as the product of a codeword optimal for a channel without CSIT and a weighting matrix dependent on the CSIT. The optimal use of CSIT is now linear precoding, which allocates power in both spatial and tem- poral dimensions. In other words, the capacity-optimal signal is zero-mean Gaussian distributed with the covariance determined by means of the precoding matrix. This optimal configuration is shown in Figure 5.
These results establish important properties of capacity- optimal signaling for a fading channel with CSIT. First, it is optimal to separate the function that exploits CSIT and the
These separation and linearity properties are the guiding prin- ciples for MIMO frequency-flat precoder designs. In particular, this article focuses on designing a precoder based on the CSIT, given predetermined channel coding and detection technique. Before discussing about specific designs, however, the structure of a system with precoding is analyzed next.
PRECODING SYSTEM STRUCTURE
The transmitter in a system with precoding consists of an encoder and a precoder, as depicted in Figure 5. The encoder intakes data bits and performs necessary coding for error correc- tion by adding redundancy, then maps the coded bits into vector symbols. The precoder processes these symbols before transmis- sion from the antennas. At the other side, the receiver decodes the noise-corrupted received signal to recover the data bits, treating the combination of the precoder and the channel as an effective channel. The structures of these processing blocks are discussed in detail next.
ENCODING STRUCTURE
An encoder contains a channel coding and interleaving block and a symbol-mapping block, delivering vector symbols to the pre- coder. We classify two broad structures for the encoder: spatial multiplexing and ST coding, based on the symbol mapping block. The spatial multiplexing structure de-multiplexes the output bits of the channel coding and interleaving block to generate inde- pendent bit streams. These bit streams are then mapped into vec- tor symbols and fed directly into the precoder, as shown in Figure 6. Since the streams are independent with individual SNR, per-stream rate adap- tation can be used.
In ST coding structure, on the other hand, the output bits of the channel coding and interleaving block are first mapped directly into symbols. These symbols are then processed by a ST encoder (such as in [38], [39]), pro- ducing vector symbols as input to the precoder, shown in Figure 7. If the ST code is capacity lossless for a channel with no CSIT (for example, the
Alamouti code for a 2×1 channel
[38]), then this structure is also capac- ity optimal for the channel with CSIT. The ST coding structure contains a single data stream; hence, only a single rate adap- tation is necessary. The rate is controlled by the FEC-code rate and the constellation design.
The difference between these two encoding structures therefore lies in the temporal dimen- sion of the symbol-level code. Spatial multiplexing spreads symbols over the spatial dimension alone,
[FIG5] An optimal configuration for exploiting CSIT.
N W^ i.i.d. Gaussian CSIT Transmitter Precoder F Encoder C X W Channel H Y Decoder +
[FIG6] A multiplexing encoding structure.
Symbol Mapping FEC Code Interleaver Symbol Mapping Input C DEMUX bk
[FIG7] A space-time (ST) encoding structure.
ST Code FEC Code Interleaver Symbol Mapping Input C bk
IEEE SIGNAL PROCESSING MAGAZINE [92] SEPTEMBER 2007
Authorized licensed use limited to: QUEENSLAND UNIVERSITY OF TECHNOLOGY. Downloaded on July 31,2010 at 16:34:04 UTC from IEEE Xplore. Restrictions apply.
Hình 3.3: Cấu trúc STBC [5, 6]3.1.2 Cấu trúc bộ tiền mã hóa tuyến tính 3.1.2 Cấu trúc bộ tiền mã hóa tuyến tính
Ta xét cấu trúc tiền mã hóa tuyến tính cho STBC. Một bộ tiền mã hóa tuyến tính có chức năng giống như một bộ tổ hợp tạo dạng và bộ tạo chùm chia nhiều chế độ cùng với sự cấp phát công suất cho từng dòng. Khai triển một ma trận tiền mã hóa:
F =UFDVF (3.1)
Hướng của các chùm trực giao là UF, trong đó mỗi cột của ma trận này biểu diễn một hướng của chùm. Công suất của mỗi các chùm trực giao làD2. Ma trận VF trộn các ký hiệu đầu vào bộ tiền mã hóa để cho vào mỗi chùm và do đó nó được đề cập đến như là ma trận tạo dạng đầu vào. Cấu trúc này được minh họa như hình 3.4.
resulting in a one-symbol-long input block, while ST coding usual- ly spreads symbols over both the spatial and the temporal dimen- sions. While these two structures have different implications on rate adaptation, this issue is not discussed in this article. Therefore, for precoding analysis and design, we will treat spatial multiplexing as a special case of ST coding with the block length of
one. Assuming a Gaussian-distributed codeword C of size N×T
with a zero mean, we define the codeword covariance matrix as
Q= TP1 E
CC∗
, (11)
where P is the transmit power (here we assume that the code- word has been scaled by the transmit power, so this definition provides the normalized covariance), and the expectation is taken over the codeword distribution. When C is spatial multi-
plexing, Q=I.
Of particular interest is ST block code (STBC), usually designed to capture the spatial diversity in the channel, assum- ing no CSIT. Diversity determines the slope of the error proba- bility versus the SNR and is related to the number of spatial links that are not fully correlated [42]. High diversity is useful in a fading link since it reduces the fade margin, which is needed to meet required link reliability. A STBC can be charac- terized by its diversity order; a full-diversity code achieves the
maximum diversity MN in a channel with Ntransmit and M
receive antennas. There is, however, a fundamental trade-off between the diversity and the multiplexing orders in ST coding [43]. The multiplexing order relates to rate-adaptation; it is the scale at which the transmission rate asymptotically increases with the SNR. A fixed-rate system therefore has a zero multi- plexing order. (Recently there has been new development of the diversity-multiplexing trade-off at finite [low] SNRs with a mod- ified definition of multiplexing order [46].) Without CSIT, STBC design achieving the optimal diversity-multiplexing trade-off is an active research area (see [44], [45] for some examples). With CSIT, on the other hand, precoding focuses on extracting a coding gain (an SNR advantage) from the CSIT; hence it is independent of, and complementary to, the diversity-multi- plexing trade-offs for ST codes.
LINEAR PRECODING STRUCTURE
The precoder is a separate transmit processing block from chan- nel and ST coding. It depends on the CSIT, but a linear precoder has a general structure. A linear precoder functions as a combi- nation of an input shaper and a multimode beamformer with per-beam power allocation. Consider the singular value decom- position (SVD) of the precoder matrix
F=UFDVF. (12)
The orthogonal beam directions are the left singular vectors UF,
of which each column represents a beam direction (pattern).
Note that UFis also the eigenvectors of the product FF∗, thus
the structure is often referred to as eigen-beamforming. The
beam power loadings are the squared singular values D2. The
right singular vectors VFmix the precoder input symbols to feed
into each beam and hence is referred to as the input shaping matrix. This structure is illustrated in Figure 8. To conserve the total transmit power, the precoder must satisfy
tr(FF∗)=1. (13)
In other words, the sum of power over all beams must be a con- stant. The individual beam power, however, can differ according to the SNR, the CSIT, and the design criterion.
Essentially, a precoder has two effects: decoupling the input signal into orthogonal spatial modes, in the form of eigen- beams, and allocating power over these beams, based on the CSIT. If the precoded, orthogonal spatial-beams match the chan-
nel eigen-directions (the eigenvectors of H∗H), there will be no
interference among signals sent on different modes, thus creat- ing parallel channels and allowing transmission of independent signal streams. This effect, however, requires the full channel knowledge at the transmitter. With partial CSIT, the precoder tries to approximately match its eigen-beams to the channel eigen-directions and therefore reduces the interference among signals sent on these beams. This is the decoupling effect. Moreover, the precoder allocates power on the beams. For orthogonal eigen-beams, if all the beams have equal power, the total radiation pattern of the transmit antenna array is isotropic. Figure 9(a) shows an example of this pattern using a uniform linear antenna array. If the beam powers are different, however, the overall transmit radiation pattern will have a specific, non- circular shape, as shown in Figure 9(b). By allocating power, the precoder effectively creates a radiation shape to match to the channel based on the CSIT, so that higher power is sent in the directions where the channel is strong and reduced or no power in the weak. More transmit antennas will increase the ability to finely shape the radiation pattern and therefore will likely to deliver more precoding gain.
[FIG8] A linear precoder structure as a multimode beamformer.
VF d1 d2 u1 u2 X C UF Σ Σ d1 d2
IEEE SIGNAL PROCESSING MAGAZINE [93] SEPTEMBER 2007
Authorized licensed use limited to: QUEENSLAND UNIVERSITY OF TECHNOLOGY. Downloaded on July 31,2010 at 16:34:04 UTC from IEEE Xplore. Restrictions apply.
Hình 3.4: Cấu trúc bộ tiền mã hóa tuyến tính [5, 6]Để bảo toàn tổng năng lượng phát, bộ tiền mã hóa phải thỏa mãn: Để bảo toàn tổng năng lượng phát, bộ tiền mã hóa phải thỏa mãn:
tr(F F∗) = 1 (3.2)
Nói cách khác, tổng công suất trên tất cả các chùm là một hằng số. Tuy nhiên, công suất của các chùm riêng lẻ là khác nhau tùy thuộc vàoSNR, SCIT và các tiêu chí thiết kế.
3.1.3 Cấu trúc thu
Xét một hệ thống với một bộ lập mã tạo nên một từ mã Cvà một bộ tiền mã hóa F tại phía phát như được mô tả trong hình 3.1. Ở phía thu, tín hiệu nhận được có dạng:
Y=HFC+N (3.3)
trong đó N là một vector của các nhiễu Gauss trắng cộng tính. Qua công thức trên, ta thấy rằng, trong hệ thống có tiền mã hóa thì HF được phía thu đối xử như là ma trận kênh hiệu dụng.
Phía thu phát hiện và giải mã tín hiệu nhận được để thu được một ước lượng của từ mã đã phátC. Phía thu có thể sử dụng một trong số các phương pháp phát hiện, dựa vào mức độ thực hiện và độ phức tạp thuật toán yêu cầu. Thuật toán giải mã trên cơ sở
MLsẽ thu được từ mã thỏa mãn:
ˆ
C= arg min
C kY−HFCk2 (3.4)
3.2 Thiết kế tiền mã hóa tối ưu
Một bộ tiền mã hóa tuyến tính bao gồm một ma trận tạo dạng đầu vào, một ma trận tạo các chùm và cấp phát năng lượng trên các chùm này. Ma trận tạo dạng đầu vào tối ưu chỉ phụ thuộc vào mã đầu vào bộ tiền mã hóa. Ma trận tạo các chùm phụ thuộc vàoCSIT và bộ cấp phát năng lượng phụ thuộc vào cả từ mã và CSIT.
Ta quan tâm đến ma trận tạo dạng đầu vào vì liên quan trực tiếp đến việc thiết kế STBC. Bộ mã hóa tạo dạng ma trận hiệp phương sai của từ mã đưa đến bộ tiền mã hóa. Bộ tiền mã hóa tương ứng chọn ma trận tạo dạng đầu vào để phù hợp với hiệp phương